Extensions of maps between graded modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:36:29Z http://mathoverflow.net/feeds/question/67040 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67040/extensions-of-maps-between-graded-modules Extensions of maps between graded modules Andreas Thom 2011-06-06T13:37:58Z 2011-06-06T13:37:58Z <p>Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\oplus n})_i,$$ which is also finitely generated free as an $R$-module. </p> <p>I am thinking for example of the image $\mathbb R[x_1,\dots,x_d]^{\oplus k}$ in $\mathbb R[x_1,\dots,x_d]^{\oplus n}$ given by a suitable $k$-tuple of homogenous vectors in $\mathbb R[x_1,\dots,x_d]^{\oplus n}$ of various degrees.</p> <blockquote> <p><strong>Question:</strong> Let $\varphi \colon N \to R^{\oplus n}$ be a homomorphism of degree $1$. Under which conditions can we extend $\varphi$ to a homomorphism $\tilde \varphi\colon R^n \to R^n$ of degree $1$?</p> </blockquote> <p>The obstruction corresponds to a class in ${\rm Ext}^1_{R}(R^{\oplus n}/N,R^{\oplus n})$ of degree zero.</p> <p>Is there any general result about classes of degree zero in such such Ext-groups over graded rings? Is there a geometric meaning of classes in graded Ext-groups, such as ${\rm Ext}^1_{R}(R^{\oplus n}/N,R)$? </p> <p>Is there a good reference for the special properties of graded Ext-groups?</p>